The decibel

Perceived levels of sound, and of other phenomena such as light and radio signals, change according to the logarithm of the actual power level. Units have been invented to take this into account.

The fundamental unit of sound change is called the decibel, abbreviated dB. A change of 1 dB is the minimum increase in sound level that you can detect, if you are expecting it. A change of - 1 dB is the minimum detectable decrease in sound volume, when you are anticipating the change. Increases in volume are positive decibel values; decreases in volume are negative values.

If you aren’t expecting the level of sound to change, then it takes about 3 dB or -3 dB of change to make a noticeable difference.

Calculating decibel values
Decibel values are calculated according to the logarithm of the ratio of change. Suppose a sound produces a power of P watts on your eardrums, and then it changes (either getting louder or softer) to a level of Q watts. The change in decibels is obtained by dividing out the ratio Q/P, taking its base-10 logarithm, and then multiplying the result by 10:

dB = 10 log (Q/P)

As an example, suppose a speaker emits 1 W of sound, and then you turn up the volume so that it emits 2 W of sound power. Then P = 1 and Q = 2, and dB = 10 log (2/1) = 10 log 2 = 10 x 0.3 = 3 dB. This is the minimum detectable level of volume change if you aren’t expecting it: a doubling of the actual sound power

If you turn the volume level back down again, then P/Q = 1/2 = 0.5, and you can calculate
dB = 10 log 0.5 = 10 × -0.3 = - 3 dB.

A change of plus or minus 10 dB is an increase or decrease in sound power of 10 times. A change of plus or minus 20 dB is a hundredfold increase or decrease in sound power. It is not unusual to encounter sounds that range in loudness over plus/minus 60 dB or more—a millionfold variation.

Sound power in terms of decibels
The above formula can be worked inside-out, so that you can determine the final sound power, given the initial sound power and the decibel change.

Suppose the initial sound power is P, and the change in decibels is dB. Let Q be the final sound power. Then Q = P antilog (dB/10).

As an example, suppose the initial power, P, is 10 W, and the change is - 3 dB. Then the final power, Q, is Q = 10 antilog (-3/10) = 10 × 0.5 = 5 W.

Decibels in real life
A typical volume control potentiometer might have a resistance range such that you can adjust the level over about plus/minus 80 dB. The audio taper ensures that the decibel increase or decrease is a straightforward function of the rotation of the shaft. Sound levels are sometimes specified in decibels relative to the threshold of hearing, or the lowest possible volume a person can detect in a quiet room, assuming their hearing is normal. This threshold is assigned the value 0 dB. Other sound levels can then be quantified, as a number of decibels such as 30 dB or 75 dB.

If a certain noise is given a loudness of 30 dB, it means it’s 30 dB above the threshold of hearing, or 1,000 times as loud as the quietest detectable noise. A noise at 60 dB is 1,000,000 times as powerful as the threshold of hearing. Sound level meters are used to determine the dB levels of various noises and acoustic environments.

A typical conversation might be at a level of about 70 dB. This is 10,000,000 times the threshold of hearing, in terms of actual sound power. The roar of the crowd at a rock concert might be 90 dB, or 1,000,000,000 times the threshold of hearing.

A sound at 100 dB, typical of the music at a large rock concert, is 10,000,000,000 times as loud, in terms of power, as a sound at the threshold of hearing. If you are sitting in the front row, and if it’s a loud band, your ears might get wallopped with peaks of 110 dB. That is 100 billion times the minimum sound power you can detect in a quiet room.